Some of my mathematical interests in the last 15 years have been focussed on geometric integration. This is a (fairly) new theory that endeavours to bring together computation and analysis of differential equations and develop discretization methods that respect underlying geometry and qualitative structure of the differential flow.

A good way to formalising geometric integration is by restricting the configuration space where the solution evolves from the entire Euclidean space to a differentiable manifold (or to its tangent bundle or a cotangent bundle).

The best (by far) resource on geometric integration is the FoCM Geometric Integration Focus Group. Other relevant websites are

• The home page of Informal year on numerical methods on manifolds, that had been organised jointly by the Cambridge numerical analysis group and the Nonlinear Centre. Although the event is long over, the relevant home page contains useful information.
• The SYNODE (SYmmetries in Numerical Ordinary Differential Equations) home page at Norwegian University of Science and Technology, Trondheim.
• A theme issue of Philosophical Transactions of Royal Society of London A on "Geometric integration: numerical solution of differential equations on manifolds" (volume 357, issue 1754), edited by Chris Budd and Arieh Iserles.
• The workshop on geometric integration at FoCM'99, organised by Chris Budd and Hans Munthe-Kaas
• The LMS Durham Symposium on "Geometric Integration", July 2000.

If you are interested in a more informal introduction to geometric integration and Lie-group methods, intended to broad scientific audience, you might download the transparancies of my Lars Onsager Lecture, "Calculating geometry" at Norwegian University of Science and Technology (October 1999). But be warned: the gzipped PostScript file is quite large!